258 research outputs found
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
We study the problem of obtaining lower bounds for polynomial calculus (PC)
and polynomial calculus resolution (PCR) on proof degree, and hence by
[Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03]
established that if the clause-variable incidence graph of a CNF formula F is a
good enough expander, then proving that F is unsatisfiable requires high PC/PCR
degree. We further develop the techniques in [AR03] to show that if one can
"cluster" clauses and variables in a way that "respects the structure" of the
formula in a certain sense, then it is sufficient that the incidence graph of
this clustered version is an expander. As a corollary of this, we prove that
the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree
when restricted to constant-degree expander graphs. This answers an open
question in [Razborov '02], and also implies that the standard CNF encoding of
the FPHP formulas require exponential proof size in polynomial calculus
resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as
shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when
restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th
Annual Computational Complexity Conference (CCC '15), June 201
Degree complexity for a modified pigeonhole principle
We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in [7]. M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov [9] and simplified by Impagliazzo, Pudlak and Sgall [8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Omega(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt [7], we obtain that the degree lower bound is tight. Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in [6]
Faster computation of the Tate pairing
This paper proposes new explicit formulas for the doubling and addition step
in Miller's algorithm to compute the Tate pairing. For Edwards curves the
formulas come from a new way of seeing the arithmetic. We state the first
geometric interpretation of the group law on Edwards curves by presenting the
functions which arise in the addition and doubling. Computing the coefficients
of the functions and the sum or double of the points is faster than with all
previously proposed formulas for pairings on Edwards curves. They are even
competitive with all published formulas for pairing computation on Weierstrass
curves. We also speed up pairing computation on Weierstrass curves in Jacobian
coordinates. Finally, we present several examples of pairing-friendly Edwards
curves.Comment: 15 pages, 2 figures. Final version accepted for publication in
Journal of Number Theor
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